Macromolecules 1992,25, 1980-1990
1980
Quantitative Theory of the Globule-to-Coil Transition. 2. Density-Density Correlation in a Globule and the Hydrodynamic Radius of a Macromolecule A. Yu. Grosberg' and D. V. Kuznetsov Institute of Chemical Physics, USSR Academy of Sciences, 117977 Moscow, USSR Received July 20, 1991; Revised Manuscript Received December 9, 1991
ABSTRACT In this series of papers new results and a brief review of the current state of mean-field theories of the condensed globular state and of the globule-to-coiltransition in the 6-regionfor a linear, homogeneous, noncharged macromolecule are presented. As a basis of our consideration,we use both Lifshitz's theory and interpolation Flory-typetheory. Complete quantitative theoretical results are obtained and compared with experimentaldata. In this paper (the second in this series) the macromolecule properties determined by the pair density-density correlation function are considered. This function for a polymer globule is calculated in the framework of mean-field Lifshitz's theory. Using the correlation function, the hydrodynamic radius of a macromolecule is found in the Kirkwood approximation. The value of fluctuationsof the macromolecule radius of gyration is calculated in the region of the globule-to-coiltransition, which is connected with a sharp change in the fluctuational regime. The existence of a maximum of fluctuations in the transition region is shown. 1. Introduction
In the previous paper' we discussed the intraglobular density distribution and corresponding experimentally observable macromolecule characteristics, such as, first of all, its radius of gyration. In the meantime, a lot of important macromolecule properties depend also on binary density-density correlations. These are, in particular, the fluctuation magnitude of the radius of gyration and the hydrodynamic radius, which is often measured in inelastic light scattering experiments. An additional interest in the chain fluctuational properties in the globular state is to account for the main difference between the globular and coil phases-this difference is just connected with the fluctuational regime (see refs 2 and 3 for more details). 2. Intraglobular Density-Density Correlation Function
As is well-known and as was mentioned in our previous paper,' the central core of a large enough globule looks like a solution of independent chains with the same density no. Therefore, the correlation function of density fluctuations in the large globule should be of the standard Ornstein-Zernike type:
Here the correlation radius t i s independent of the globule size and coincides with the correlation length of the polymer solution with no concentration; Le., F = (u2/6)(C/B2) (see refs 4 and 5 and Appendices B and C). As usual, u stands for the link's size, and B and C are the second and third quasimonomer virial coefficients. The expression in eq 1is valid for a large globule core only but not for a globule surface layer. Therefore, its applicability breaks down in the globule-to-coil transition region. For the globule at an arbitrary temperature, including the globule-to-coil transition point, the intraglobular correlator can be obtained as the static response function using the following general statistical theorem:'
Here 6n(X) 5 nr(X) - n(X), nr(X) is an instant value of the polymer link density in the X point, the smoothed link density n(X) (nr(X)) = SF/Sq, and the broken brackets mean an average over the thermodynamical ensemble of chains. p(X) stands for the effectivepotential, which is a conjugate to the density n(X). Since the free energy functional F{cp] of the globular chain under the action of an external field p(X) is just the key construction of Lifshitz-type globule theory? one may derive the equation for the correlation function using this Lifshitztype approach. The derivation is performed in Appendix A; however, the resulting eq A9 is rather tedious. Its analysis in the volume approximation (Appendix B)yields eq 1. Some simplifications can be reached using dimensionless variables (Appendix C). We discuss separately the results which can be obtained for different values of physical interest: the fluctuation magnitude of the macromolecule radius of gyration (section 3) and the hydrodynamic radius (section 4).
3. Magnitude of t h e Fluctuations of the Macromolecule Radius of Gyration 3.1. Lifshitz-Type Approach for Globule Fluctuations. We characterize fluctuations of the globule gyration radius with the value [(Sr*) - (Sr2)21/(Sr2)2, where the instantaneous radius of gyration equals Sr = [(l/N)Jnr(X) x 2 dXI1l2,where X is the radius-vector from the globule's center a n d N = Jnr(X) dX is the total number of monomers per chain. Since
0024-9297/92/2225-1980$03.00/00 1992 American Chemical Society
Quantitative Theory of Globule-to-CoilTransition. 2 1981
Macromolecules, Vol. 25, No. 7, 1992
-.
and
as
= N'J(nr(X1) nr(XJ)x,2x,2 dX1 dX2 (3b)
the difference between these two values is expressed through the pair correlation function CP(X1,Xz) (eq 2):
-
( ~ 4 ) ( ~ 2 ) '
(fW
- JJ9(Xl,Xz)
~ 1 2 x 2 2dXl dX2
[Jn(X) x 2 dX1'
(4)
t
-
where the smoothed density spatial distribution n(X) E (nr(X)) has been calculated in the previous paper.' Analysis of the expression in eq 4 is essentially simpler than that of the correlation function CP itself. Nevertheless, this analysis is also rather complex; we put it into Appendix D. Here we formulate the following main results: (i) The dimensionlessversion of eq 4 can be written in the form
where the ratio &/a3 is a characteristic of the polymer chain rigidity.' s ( t ) is the complex integral expression (see Appendix D), but it depends on the single value of the reduced temperature only:
(Tstands for temperature). The expression in eq 5 proves, in particular, the fact that the globule fluctuation magnitude decreases with an increase in the polymer rigidity (i.e., with a decrease of the a l a 3 parameter). On the qualitative level this fact was formulated in refs 2 and 3. (ii) The poor solvent asymptotic behavior when -t >> 1 is s ( t ) N 5.6t-2. (iii) We have calculated numerically s ( t ) function behavior in the range of smaller -t near the transition point; the result is shown in Figure 1. As wae expected, the s ( t ) function grows sharply, when the reduced temperature approaches the transition point, and in this very point it reaches the value s ( t 3 S! 3.5, so that
(7)
At the same time in the ideal Gaussian coil, according to ref 7
Therefore, the fluctuation magnitude in the transition point is greater than that in the Gaussian state for the chain with &/a3 >0.08.This is the case for many flexible polymers. For example, according to our estimations8 and to those expressed in refs 9 and 10, for polystyrene f i l a 3 parameter equals approximately 0.1 + 0.2. 3.2. Chain Fluctuations in the Transition Region. Our approach based on Lifshitz's globule theory cannot be applied to the coil state and cannot describe by itself, without modifications, the whole transition area. In ref 1 we have shown that for rigid chains a two-level system approximation is valid for calculation of the mean-square
0
Figure 1. Fluctuation values of the radius of gyration of a polymer globule scaledby (*/a3)-1vs the t parameter of reduced temperature (t N N f 2 ( T- @/e) near the transition point to the coil state. The corresponding value of the expansion factor is shown at the bottom of this figure. The dashed lines are the results of volume approximation. gyration radius (5"). Using this approximation, we have found the simple interpolation expression for ( S2). The same approximation leads to an analogous expression for (S4), and as a result we have the following formula for fluctuations:
where 8' (S,l,b2)/ (Smile); ( SglobZ)and (Scoilz) are macromolecularmean-squareradii of gyration in globular and coil states, respectively. The difference between free energies of the globule and the coil F was calculated in ref 1. The last term in the righthand side of this expression corresponds to macromolecular fluctuations between different states (levels), globular and coil, and it is this very term which leads to the appearance of a maximum of fluctuations in the transition region. The interpolation expression in eq 9 is valid for rigid chains only, because just for that case we expect the bimodal distribution in the transition region,' and this very form of the distribution is the basis of the two-level approximation. 3.3. Flory-Type Interpolation Theory for Chain Fluctuations. To analyze the fluctuations of the macromolecular radius of gyration in the region of the globulecoil transition, we can also use the Flory-type interpolation approach, developed in ref 10 and already discussed in the previous paper.' We reiterate that this approach is based on treating the free energy of a polymer-solvent system as a whole interpolation expression of one variable only-the expansion factor a of the gyration radius S (eq
1982 Grosberg and Kuznetaov
Macromolecules, Vol. 25, No. 7, 1992
23 in ref 1):
-* 0.5
0.4
where in accordance with the discussion in ref 1 the parameters are chosen as follows: u
= 0.2836(&/~~)'/~, w 2 3.5454(&/a3I2
(11) The mean values of the expansion factor can be calculated from the definition
0.3
0.2
I 0.1
Using the saddle-point approximation, i.e., the Taylor expansion of the F(a)function near its minimum point, one can obtain the following simple result:
(S4)- (s2)2 - ( ( a 2- (a2H2) (S2)2 (a2)2 q 3 - at2 + 3wa0-6)-' (12) 9 Here the "saddle-point" value a0 is defined from the condition of the free energy minimum a; - a. = U t , f f
-20 1 .o
I I I
+.it
-10
9
0.8
0.6
0.4
+ Wac3
and it is, of course, just the equilibrium value of a (eq 24 in ref 1). The result of eq 12 is valid, at least qualitatively, for a flexible enough chain only, namely, for the case where w > 4 / 7 2 ~or f i l a 3 > 0.04. The righthand side of eq 12 has a nonphysical singularity in the opposite case, when the globule-to-coil transition looks like the first-order phase transition and when the Flory-type approach predicts the discontinuous dependence of the a0 value on reduced temperature teff. For flexible macromolecules with &/a3 < 0.04 the expression in eq 12 predicts the maximum of fluctuations at the reduced temperature point tmm flu& which can be determined from the equation aO(tmar fluct) = ( 9 ~ ) ' ~2.377(&/a3)'I2 ~
The result is illustrated in Figure 2, where the dependence of eq 12 and the corresponding dependence of a0 on t e f f are shown for &/a3 = 0.1. 3.4. Discussion. Thus, we have arrived at two interpolation formulas for the description of the fluctuation magnitude of the radius of gyration in the whole region of the globule-coil transition: the expression in eq 9 is correct for rigid polymer chains (approximately &/a3 S 0.04), and the expression in eq 12 is valid for flexible chains (&/a3 3 0.04). The difference between forms of these two equations reflects the difference in the physical picture of the transition:' the bimodal distribution function and the corresponding idea of a two-level system are inherent for rigid chains; the more smooth type of transition with the monomodal distribution is the case for flexible chains. The great fluctuations in the rigid chains correspond physically to the spontaneous jumps between the globular and coil states; it is just the typical mechanism of the so-called heterophase fluctuations. On the other hand, great fluctuations in a flexible polymer are caused by a growth in the width of the single peak of the monomodal distribution; this mechanism is analogous to the one for
0.2
Figure 2. Dependences of fluctuation values of the macromolecule radius of gyration and ita corresponding expansion factor vs the teff parameter of the effective reduced temperature in the
globule-to-coil transition region. These results were calculated using the Flory-type theory when &/a3 = 0.1. the fluctuations near the critical point or the second-order phase transition. In spite of that essential difference in the physical picture, the qualitative behavior of the fluctuation magnitude is practically the same in both these cases. As is seen in Figures 1 and 2, both approaches yield quite close results for the fluctuation magnitude in the globular region and predict the presence of a maximum of fluctuations in the transition region. Of course, the existence of this maximum could be expected: whatever the true mechanism (order) of this transition might be, the function of the system distribution in phase space expands near it; i.e., fluctuations get access to states of rather different sizes (close to the coil and globular states, respectively), and the magnitude of the fluctuations reaches its maximum. Such a maximum was observed in a real experiment" and in computer simulation.12
4. Hydrodynamic Radius of a Macromolecule In experiments with isolated macromoleculessuspended in dilute solution, not only the radius of gyration but also another parameter of expansion-the so-called hydrodynamic radius & erausually measured. RH can be measured in viscosimetric and in inelastic scattering experiments because the very size of the macromolecule characterizes its friction when it moves through the medium of a liquid solvent. If the diffusion coefficient of the macromolecule equals D and, correspondingly, its friction coefficientisf = TID,then its hydrodynamic radius can be defined via the relation
D = (T/~TW)(RH-')
(13) where 7 stands for the solvent viscosity. Due to the well-
Macromolecules, Vol. 25, No. 7, 1992
Quantitative Theory of Globule-to-Coil Transition. 2 1983
known phenomenon of hydrodynamic interactions, the friction of the chain monomers is nonadditive; i.e., the value off = T / D RH is not proportional to the chain length N. For the polymer coil the hydrodynamic radius RHscales as the gyration radius; i.e., it is proportional to hn/2 for the Gaussian coil and to Nv for the swollen c0il.13 These facta are usually described by the Kirkwood-Riseman theory,13which leads for RHto a relation of the type
-
This expression is often considered to be the exact definition of the hydrodynamic radius. However, when the physical definition (eq 13) is used, then the KirkwoodRiseman formula in eq 14 becomes rough and ita accuracy is on the order of a slowly changing constant numerical coefficient.14 The Kirkwood-Riseman theory is based on the supposition that all monomers are equally bathed by the solvent. Naturally, it is not the case for anon-free-draining chain, especially for a dense globular chain. For example, if the density nr(X)is uniformly distributed in a sphere of radius R and it does not fluctuate, then the KirkwoodRiseman expressiongives RH= (5/6)R instead of the correct result for a solid sphere, RH = R. This coefficient 5/6 # 1characterizes the accuracy of the Kirkwood-Riseman theory. Nevertheless, we use this theory and the expression in eq 14-because we simply do not know what would be better. At the same time we are ready beforehand to get '/6 or 17%. an accuracy on the order of 1 - 5/6 4.1. Hydrodynamic Radius of a Slightly Compressed Coil. For the Gaussian coil the Kirkwood-Riseman expression gives, as is well-known (see, for example, ref 15)
-40
-20
ttr
0
- 40
-20
ttr
0
Figure 3. Functions hl(t)and hz(t)determining the expansion factor of the polymer globule hydrodynamicradius (eq 16). The h1 function was calculated using the smoothed density distribution, while the hz function was calculated using the density pair correlation function. When */a3
